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The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost.

In this paper, a feedback control law based on two time‐scale assignment and state extension is implemented for the nonlinear non‐minimum phase dynamics of three degree‐of‐freedom autonomo...

The problem of interval estimation of the time delay of the coupling between oscillatory systems from observed time series is considered. It is shown that the known asymptotic estimates based on the empirical model in the form of a system of first-order phase oscillators and the maximum likelihood formalism can lead to false inferences of the value of the time delay in two typical situations: 1) nonlinear low-dimensional systems whose phases are well-determined but, as a result of significant amplitude fluctuations, the phase approximation is insufficient for describing the dynamics and 2) systems whose phases are defined not quite well because of too large amplitude fluctuations. A method for empirical diagnostics of problematic situations and its modification (coarse estimation) providing a low probability of false inferences in these situation are proposed. The efficiency of the diagnostic criterion and coarse estimation suggested is demonstrated on reference systems with different dynamic properties (linear stochastic oscillators, van der Pol oscillators, and Ressler and Lorenz chaotic systems).

Josephson junctions with weak-links of exotic materials allow the elucidation of the Josephson effect in previously unexplored regimes. Further, such devices offer a direct probe of novel material properties, for example in the search for Majorana fermions. In this work, we report on DC and AC Josephson effect of high-mobility, hexagonal boron nitride (h-BN) encapsulated graphene Josephson junctions. On the application of RF radiation, we measure phase-locked Shapiro steps. An unexpected bistability between $\pm 1$ steps is observed with switching times on the order of seconds. A critical scaling of a bistable state is measured directly from the switching time, allowing for direct comparison to numerical simulations. We show such intermittent chaotic behavior is a consequence of the nonlinear dynamics of the junction and has a sensitive dependence on the current-phase relation. This work draws connections between nonlinear phenomena in dynamical systems and their implications for ongoing condensed matter experiments exploring topology and exotic physics.

We investigate nonlinear phase dynamics of an ideal kink mode, induced by E × B flow. Here the phase is the cross phase (θ c) between perturbed stream function of velocity ( ϕ ˜ ) and magnetic field ( ψ ˜ ), i.e. θ c = θ ϕ − θ ψ . A dimensionless parameter, analogous to the Richardson number, R i = 16 γ kink 2 / ω ˆ E 2 (γ kink: the normalized growth rate of the pure kink mode; ω ˆ E : normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear. When R i > 1, θ c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R i ≤ 1, θ c enters a phase slipping or oscillating state, corresponding to a nonmodal solution. The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.

We examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.

A minimal model of nonlinear phase dynamics in drift waves is shown to support phase bore solutions. Coupled nonlinear equations for amplitude, phase, and zonal flow are derived for the Hasegawa-Mima system and specialized to the case of spatiotemporally constant amplitude. In that limit, phase curvature (finite second derivative of the phase with respect to the radius) alone generates propagating shear flows. The phase field evolves nonlinearly by a competition between phase steepening and dispersion. The analytical solution of the model reveals that the phase bore solutions so obtained realize the concept of a phase slip in a concrete dynamical model of drift wave dynamics. The implications for phase turbulence are discussed.

We study theoretically the non linear polarization dynamics of Vertical-Cavity Surface-Emitting Lasers in the presence of an external cavity providing delayed optical feedback and cross polarization re-injection. We show that far from the laser threshold, the dynamics remains confined close to the equatorial plane of a Stokes sphere of a given radius and we reduce the dynamics to a dynamical system composed of two phases: the orientation phase of the quasi-linear polarization and the optical phase of the field. We explore the complex modal structure given by the double feedback configuration and recovers as particular cases the Lang-Kobayashi modes and the modes founds by Giudici et al. [1]. We also re-interpret the square waves switching dynamics as phase kinks.

The simplified models of neural networks based on biophysical Hodgkin–Huxley neurons are studied with a focus on coherent-phase dynamics. In our approach, each neuron is considered as a nonlinear oscillator, and collective dynamics of a mesoscopic network of neural oscillators are studied using the methods of nonlinear dynamics. We explore the mechanisms for synchrony, clustering and their breakup in the synaptic parameter space and discuss implications to temporal aspects of neural-information processing.

We study the nonlinear phase equation for propagating patterns. We investigate the transition from a propagating localized pattern to a space-filling spatiotemporally disordered pattern and discuss in detail to what extent there are propagating localized states that breathe in time periodically, quasiperiodically, and chaotically. Differences and similarities to the phenomena occurring for the quintic complex Ginzburg-Landau equation are elucidated. We also discuss for which experimentally accessible systems one could observe the phenomena described.

We discuss the phase dynamics in the system of a set of nonlinear coupled oscillators. We find a direct and simple geometric method to observe the geometric phase for this system, and it provides a possibility for the experimental measurements.

An unusual bifurcation sequence exhibited in a recently proposed nonlinear delay electro-optic phase dynamics setup, is explored both experimentally and analytically. Strong agreements indicate that well-controlled dynamical complexity can be usefull in many emerging applications.

We generalize our recent approach to reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E, 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from the multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. The partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling., Comment: 6 pages, 5 figures, 27 references

Improving the frequency precision by synchronizing a lattice of oscillators is studied in the phase reduction limit. For the most commonly studied case of purely dissipative phase coupling (the Kuramoto model) I confirm that the frequency precision of N oscillators perturbed by independent noise sources is improved by a factor N as expected from simple averaging arguments. In the presence of reactive coupling, such as will typically be the case for non-dissipatively coupled oscillators based on high-Q resonators, the synchronized state consists of target like waves radiating from a local source which is a region of higher frequency oscillators. In this state all the oscillators evolve with the same frequency, however I show that the improvement of the frequency precision is independent of N for large N, but instead depends on the disorder and reflects the dependence of the frequency of the synchronized state on just those oscillators in the source region of the waves., Comment: 18 pages, 8 figures; clarified wording of text and added references

A crucial insight into handwriting dynamics is embodied in the idea that stable, robust handwriting movements correspond to attractors of an oscillatory dynamical system. We present a phase dynamic model of visuomotor performance involved in copying simple oriented lines. Our studies on human performance in copying oriented lines revealed a systematic error pattern in orientation of drawn lines, i.e., lines at certain orientation are drawn more accurately than at other values. Furthermore, human subjects exhibit "flips" in direction at certain characteristic orientations. It is argued that this flipping behavior has its roots in the fact that copying process is inherently ambiguous-a line of given orientation may be drawn in two different (mutually opposite) directions producing the same end result. The systematic error patterns seen in human copying performance is probably a result of the attempt of our visuomotor system to cope with this ambiguity and still be able to produce accurate copying movements. The proposed nonlinear phase-dynamic model explains the experimentally observed copying error pattern and also the flipping behavior with remarkable accuracy. � 2009 Springer-Verlag.

This work concerns output trajectory tracking for the McDonnell Dougglas YAV68B Harrier VSTOL aircraft. This system can perform three distinct flight modes, and is modelled with a Lagrangian nonlinear hybrid non-minimum phase dynamics. Output tracking is solved using classical input/output linearization scheme associated to an internal dynamics stable inversion for each flight mode dynamics. The performance of the global switched control law is discussed and validated through simulation tests.

[1] Based on the nonlinear techniques for estimation of coupling between oscillatory systems, we investigate the dynamics of climatic modes characterizing global and the Northern Hemisphere (NH) processes. In particular, indices of the North Atlantic Oscillation (NAO) and El Nino–Southern Oscillation (ENSO) for 1950–2004 are analyzed. The methods based on phase dynamics modeling and nonlinear “Granger causality” are used. We infer that ENSO affects NAO during the last half a century with confidence probability higher than 0.95. The influence is characterized with time delay in the range from a couple of months up to three years and increases during the last decade.

We experimentally address long-time dynamics of an artificially curved growth interface in directional solidification. Repetitive cell nucleations are found to appear in a disordered way, but to eventually organize themselves in a coherent way, for long times. This behavior is recovered by simulation of a nonlinear advection-diffusion model for phase dynamics. The existence of a periodic attractor is supported by the derivation of a Lyapunov functional for this model.

We consider the phase dynamics of an ensemble of Kuramoto oscillators whose eigenfrequencies are perturbed to model the openness of living systems, and we show that it exhibits time-localized epochs of synchrony. A new quantitative measure is used to show that the model compares well with electroencephalography data recorded from a healthy awake human.

Recent advances in subpicosecond generation and amplification with semiconductor lasers have shown the necessity of extending and modifying existing theoretical models1 to include phenomena influencing subpicosecond pulse amplification. To model subpicosecond pulse evolution in semiconductor laser amplifiers (SLA), spectral dependence of both real and imaginary parts of the complex susceptibility (index and gain) within the pulse bandwidth must be included. In addition to the well known gain saturation and selfphase modulation (SPM) due to carrier depletion1, it is necessary to account for the recently observed rapid gain and phase dynamics which have been attributed to carrier heating2. The effects of an instantaneous nonlinear index n2 must also be included.

In this note we will introduce in the first section the concept of phase dynamics, followed by an outline of some recent developments associated with the properties of nonlinear phase equations, and we will conclude with an extensive list of references on phase dynamics and closely related subjects including envelope equations, the motion of defects, and the coupling between amplitude and phase equations.

We present the nonlinear phase equations describing the stability of a one-dimensional periodic pattern, with mean flow effect due to Galilean invariance. We show that the phase dynamics is second order in time, and could lead to an oscillatory instability of the pattern.